Problem: Simplify the following expression: $y = \dfrac{9x^2- 35x+24}{x - 3}$
Answer: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(24)} &=& 216 \\ {a} + {b} &=& &=& {-35} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $216$ and add them together. The factors that add up to ${-35}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-8}$ and ${b}$ is ${-27}$ $ \begin{eqnarray} {ab} &=& ({-8})({-27}) &=& 216 \\ {a} + {b} &=& {-8} + {-27} &=& -35 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({9}x^2 {-8}x) + ({-27}x +{24}) $ Factor out the common factors: $ x(9x - 8) - 3(9x - 8)$ Now factor out $(9x - 8)$ $ (9x - 8)(x - 3)$ The original expression can therefore be written: $ \dfrac{(9x - 8)(x - 3)}{x - 3}$ We are dividing by $x - 3$ , so $x - 3 \neq 0$ Therefore, $x \neq 3$ This leaves us with $9x - 8; x \neq 3$.